I suggest that you add a series of "end game" quests which would not only serve to introduce the dragon civilization but also provide players with a counter. The main purpose of these quests would be to introduce a new hero to the players which would be based on their race.

For example:

Humans on completion of the series of quests would be rewarded with a griffon mounted hero (same stats as regular hero except: loses aoe heal and aoe buff/regen, gains talon smash (aoe pushback/damage) and flight)

the orc/human and elf heros would be armed with a long lance, the dwarf a dragon hunting harpoon (short range)

These heros would be the race's dragon hunters, the main purpose would be to track and chase the dragons around the battle field. they should inflict massive damage on dragons when they catch up to them but be less effective against regular ground troops than the regular hero. this would add a hard counter to the dragons and a new dynamic to the battle while keeping the dragons feeling as powerful as they should be and adding instead of subtracting to the game experience.

I like the idea about the quest and the secondary hero, but to be honest - I don`t think we`ll get to include this any time soon.
However - I do agree - we need a hard counter against the Dragons... what I had in mind is making the Player hero the Dragon Slayer unit you talked about, all we`d have to do is give him x6 damage against Dragons, and should be good to go.